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power

This entry is about the formal dual to tensoring in the generality of category theory. For the different concept of cotensor product of comodules see there.


Contents

Idea

In a closed symmetric monoidal category VV the internal hom [,]:V op×VV[-,-] : V^{op} \times V \to V satisfies the natural isomorphism

[v 1,[v 2,v 3]][v 2,[v 1,v 3]] [v_1,[v_2,v_3]] \simeq [v_2,[v_1,v_3]]

for all objects v iVv_i \in V. If we regard VV as a VV-enriched category we write V(v 1,v 2):=[v 1,v 2]V(v_1,v_2) := [v_1,v_2] and this reads

V(v 1,V(v 2,v 3))V(v 2,V(v 1,v 3)). V(v_1,V(v_2,v_3)) \simeq V(v_2,V(v_1,v_3)) \,.

If we now pass more generally to any VV-enriched category CC then we still have the enriched hom object functor C(,):C op×CVC(-,-) : C^{op} \times C \to V. One says that CC is powered over VV if it is in addition equipped also with a mixed operation :V op×CC\pitchfork : V^{op} \times C \to C such that (v,c)\pitchfork(v,c) behaves as if it were a hom of the object vVv \in V into the object cCc \in C in that it satisfies the natural isomorphism

C(c 1,(v,c 2))V(v,C(c 1,c 2)). C(c_1,\pitchfork(v,c_2)) \simeq V(v,C(c_1,c_2)) \,.

Definition

Definition

Let VV be a closed symmetric monoidal category. In a VV-enriched category CC, the power of an object yCy\in C by an object vVv\in V is an object (v,y)C\pitchfork(v,y) \in C with a natural isomorphism

C(x,(v,y))V(v,C(x,y)) C(x, \pitchfork(v,y)) \cong V(v, C(x,y))

where C(,)C(-,-) is the VV-valued hom of CC and V(,)V(-,-) is the internal hom of VV.

We say that CC is powered or cotensored over VV if all such power objects exist.

Remark

Powers are frequently called cotensors and a VV-category having all powers is called cotensored, while the word “power” is reserved for the case V=V= Set. However, there seems to be no good reason for making this distinction. Moreover, the word “tensor” is fairly overused, and unfortunate since a tensor (= a copower) is a colimit, while a cotensor (= power) is a limit.

Properties

  • Powers are a special sort of weighted limits. Conversely, all weighted limits can be constructed from powers together with conical limits. The dual colimit notion of a power is a copower.

Examples

  • VV itself is always powered over itself, with (v 1,v 2):=[v 1,v 2]\pitchfork(v_1,v_2) := [v_1,v_2].

  • Every locally small category CC (V=(Set,×)V = (Set,\times) ) with all products is powered over Set: the powering operation

    (S,c):= sSc \pitchfork(S,c) := \prod_{s\in S} c

    of an object cc by a set SS forms the |S||S|-fold cartesian product of cc with itself, where |S||S| is the cardinality of SS.

    The defining natural isomorphism

    Hom C(c 1,(S,c 2))Hom Set(S,Hom C(c 1,c 2)) Hom_C(c_1,\pitchfork(S,c_2))\simeq Hom_{Set}(S,Hom_C(c_1,c_2))

    is effectively the definition of the product (see limit).

References

Section 3.7 of

Section 6.5 of

  • Francis Borceux, Handbook of categorical algebra, vol. 2

Revised on May 12, 2016 03:00:13 by Urs Schreiber (131.220.184.222)